- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤ

Values

[.,.] => [1] => 1

[.,[.,.]] => [2,1] => 2

[[.,.],.] => [1,2] => 1

[.,[.,[.,.]]] => [3,2,1] => 3

[.,[[.,.],.]] => [2,3,1] => 2

[[.,.],[.,.]] => [1,3,2] => 1

[[.,[.,.]],.] => [2,1,3] => 2

[[[.,.],.],.] => [1,2,3] => 1

[.,[.,[.,[.,.]]]] => [4,3,2,1] => 4

[.,[.,[[.,.],.]]] => [3,4,2,1] => 3

[.,[[.,.],[.,.]]] => [2,4,3,1] => 2

[.,[[.,[.,.]],.]] => [3,2,4,1] => 3

[.,[[[.,.],.],.]] => [2,3,4,1] => 2

[[.,.],[.,[.,.]]] => [1,4,3,2] => 1

[[.,.],[[.,.],.]] => [1,3,4,2] => 1

[[.,[.,.]],[.,.]] => [2,1,4,3] => 2

[[[.,.],.],[.,.]] => [1,2,4,3] => 1

[[.,[.,[.,.]]],.] => [3,2,1,4] => 3

[[.,[[.,.],.]],.] => [2,3,1,4] => 2

[[[.,.],[.,.]],.] => [1,3,2,4] => 1

[[[.,[.,.]],.],.] => [2,1,3,4] => 2

[[[[.,.],.],.],.] => [1,2,3,4] => 1

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 5

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 4

[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 3

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 4

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 3

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 2

[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 2

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 3

[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 2

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 4

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 3

[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 2

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 3

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 2

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 1

[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 2

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 1

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 1

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 3

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 2

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 1

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 2

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 1

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 4

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 3

[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 2

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 3

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 2

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 1

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 1

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 2

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 1

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 3

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 1

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 2

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 6

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 5

[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 4

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 5

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 4

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 3

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 3

[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 4

[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 3

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 5

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 4

[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 3

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 4

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 3

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 2

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 2

[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 2

[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 2

[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 2

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 3

[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 3

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 2

[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 2

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 4

[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 3

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 2

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 3

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 5

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 4

[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 3

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 4

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 3

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 2

[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 2

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 3

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 2

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Description

The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$

Map

to 312-avoiding permutation

Description

Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.